Why won't Evee open the selected material in compositing
You are mixing up some things here:

The node-based material definition process is not the same thing as "compositing". They share the feature of being node-based but they are not the same thing. 

Since they are two separate things, Blender 2.8 split the editor into two: the "Compositing" editor and the "Shader Editor" (Eevee doesn't really matter here, this is an unrelated change).



The one that you find in the "Compositing" workspace is the compositing editor. The one you need is in the "Shading" workspace. 



It is also in whatever workspace you want, if you select the editor type to be "Shader Editor".